Let $\mathrm { z } = \mathrm { a } + i b , \mathrm { b } \neq 0$ be complex numbers satisfying $\mathrm { z } ^ { 2 } = \overline { \mathrm { z } } \cdot 2 ^ { 1 - | z | }$. Then the least value of $n \in N$, such that $z ^ { n } = ( z + 1 ) ^ { n }$, is equal to $\_\_\_\_$.